Tight bound for independent domination of cubic graphs without $4$-cycles

Abstract

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $γ(G)$, is the minimum size of a dominating set of $G$. The independent domination number of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent. Recently, Abrishami and Henning proved that if $G$ is a cubic graph with girth at least $6$, then $i(G) \le \frac{4}{11}|V(G)|$. We show a result that not only improves upon the upper bound of the aforementioned result, but also applies to a larger class of graphs, and is also tight. Namely, we prove that if $G$ is a cubic graph without $4$-cycles, then $i(G) \le \frac{5}{14}|V(G)|$, which is tight. Our result also implies that every cubic graph $G$ without $4$-cycles satisfies $\frac{i(G)}{γ(G)} \le \frac{5}{4}$, which partially answers a question by O and West in the affirmative.

Figures (4)

• Figure 1: Graphs whose independent domination number achieves a tight upper bound
• Figure 2: The graphs $T_{6,0}$ and $T_{3,2}$
• Figure 3: Illustrations for the proof of Claim \ref{['claim:n1']}
• Figure 4: An illustration for the cycle $C$

Theorems & Definitions (25)

• Theorem 1.1: cho2021independent
• Conjecture 1.2: see goddard2013independent
• Conjecture 1.3: goddard2012independent
• Theorem 1.4: abrishami2018independent
• Theorem 1.5
• Theorem 1.6
• Theorem 1.8
• proof
• Claim 2.1
• proof